/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 65 $$ \text { For Problems 45-68,... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 65

$$ \text { For Problems 45-68, solve each equation. (Objective 2) } $$ $$ 64 x^{2}=81 $$

Short Answer

Expert verified
The solutions are \( x = \frac{9}{8} \) and \( x = -\frac{9}{8} \).

Step by step solution

01

Isolate the Squared Term

First, we start by isolating the squared term in the equation. We want to have the term containing the square on one side of the equation: \[ 64x^2 = 81 \] This step is already complete, as the squared term \( x^2 \) with coefficient 64 is by itself on one side of the equation.
02

Divide to Simplify

Next, divide both sides of the equation by 64 to simplify the equation and solve for the square of \( x \): \[ x^2 = \frac{81}{64} \]
03

Solve for x

To solve for \( x \), take the square root of both sides of the equation. Remember that taking the square root introduces both positive and negative solutions: \[ x = \pm \sqrt{\frac{81}{64}} \] This can be simplified further: \[ x = \pm \frac{\sqrt{81}}{\sqrt{64}} \] \[ x = \pm \frac{9}{8} \] Therefore, \( x = \frac{9}{8} \) or \( x = -\frac{9}{8} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
When solving quadratic equations, like the one in our example, isolating the variable refers to getting the term with the variable by itself on one side of the equation. In our problem, the variable is part of a squared term \( x^2 \) with a coefficient of 64: \[ 64x^2 = 81 \]To isolate \( x^2 \), we need to move the constant 64 to the other side. We do this by dividing both sides of the equation by 64. This transforms the equation to: \[ x^2 = \frac{81}{64} \]By isolating the squared term, we're one step closer to solving for \( x \). This is a common first step in solving these types of equations and helps simplify the problem.
Square Root Property
After isolating the squared term, we use the square root property to solve for \( x \). This property allows us to find the value of a variable when it is squared. Given the equation: \[ x^2 = \frac{81}{64} \] Taking the square root of both sides helps remove the square from the \( x^2 \): \[ x = \pm \sqrt{\frac{81}{64}} \] The square root property introduces two solutions: one positive and one negative. Hence, we write \( x = \pm \). This acknowledges that both \( x = \frac{9}{8} \) and \( x = -\frac{9}{8} \) satisfy the equation. Thus, it's essential to check both possible solutions.
Simplifying Fractions
Simplifying fractions is often necessary to find the most reduced form of an answer. After taking the square root of both sides in our equation, we're left with: \[ x = \pm \sqrt{\frac{81}{64}} \]This expression can be simplified by separating the square root across the fraction: \[ x = \pm \frac{\sqrt{81}}{\sqrt{64}} \]Both 81 and 64 are perfect squares, so we can find their square roots: - \( \sqrt{81} = 9 \) - \( \sqrt{64} = 8 \)Thus, the fraction simplifies to: \[ x = \pm \frac{9}{8} \]This step ensures that our solution is not only correct but also presented in its simplest form, making it easier to interpret and use in further calculations.

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